Question

Suppose that p>1 has the property that for all integers a, b, if p divides the product ab, then p divides a or p divides b.
Show that p is a prime.
Please, help

Answer 1

@rational

Answer 2

Suppose \(p = d_1*d_2\) is composite with the given property.

Next consider two integers \(a=d_1\) and \(b=d_2\),
clearly \(\large p\) divides \(\large a*b\)
However \(\large p\) cannot divide \(\large a\) or \(\large b\) individually because \(\large p \gt a\) and \(\large p \gt b\)

So \(\large p\) is not composite.

Answer 3

Is it so particular?? How about the case like 6 | 72 where a =36 and b =2,
but 6 is not a prime.

Answer 4

@rational

Answer 5

The proof is about proving that the given property will not hold for a composite number like, 6.

Answer 6

:) , let me think more, still not digest it.

Answer 7

The given property states below :
Suppose that p>1 has the property that `for all integers a, b,` if p divides the product ab, then p divides a or p divides b.

Answer 8

what we have shown in the proof above is that, there will be some integers `a,b` for which the given property will not hold for composite numbers, like 6

Answer 9

Oh, yes, got it so far.

Answer 10

for example :

6|3*2

but 6 does not divide 3
6 does not divide 2

failing the given property

Answer 11

I can show that every composite number fails the given property using above trick ^^

Answer 12

hihihi. thank you.